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A353774 Expansion of e.g.f. 1/(1 - (exp(x) - 1)^3).

%I A353774 #19 May 09 2022 08:38:28
%S A353774 1,0,0,6,36,150,1260,16926,197316,2286150,32821020,548528046,
%T A353774 9515702196,174531124950,3521913283980,76969474578366,
%U A353774 1777400236160676,43405229295464550,1126972561394470140,30949983774936839886,893095888222540548756,27035433957000465352950
%N A353774 Expansion of e.g.f. 1/(1 - (exp(x) - 1)^3).
%H A353774 Seiichi Manyama, <a href="/A353774/b353774.txt">Table of n, a(n) for n = 0..424</a>
%F A353774 G.f.: Sum_{k>=0} (3*k)! * x^(3*k)/Product_{j=1..3*k} (1 - j * x).
%F A353774 a(0) = 1; a(n) = 6 * Sum_{k=1..n} binomial(n,k) * Stirling2(k,3) * a(n-k).
%F A353774 a(n) = Sum_{k=0..floor(n/3)} (3*k)! * Stirling2(n,3*k).
%F A353774 a(n) ~ n! / (6 * log(2)^(n+1)). - _Vaclav Kotesovec_, May 08 2022
%o A353774 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-(exp(x)-1)^3)))
%o A353774 (PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (3*k)!*x^(3*k)/prod(j=1, 3*k, 1-j*x)))
%o A353774 (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=6*sum(j=1, i, binomial(i, j)*stirling(j, 3, 2)*v[i-j+1])); v;
%o A353774 (PARI) a(n) = sum(k=0, n\3, (3*k)!*stirling(n, 3*k, 2));
%Y A353774 Cf. A000670, A052841, A353775.
%Y A353774 Cf. A143815, A346894, A353118, A353664.
%K A353774 nonn
%O A353774 0,4
%A A353774 _Seiichi Manyama_, May 07 2022